In a large pot, melt the butter over medium-high heat. Add the onion and cook, stirring frequently with a wooden spoon, until translucent and soft, about 2 minutes. Add the garlic and cook for 1 minute. Continue to stir frequently. Add the mustard and stir to combine.

Stir in the sweet rice flour until a paste forms. Cook, stirring constantly, until the paste is a light golden brown, about 4 minutes. Whisk in the chicken stock in a slow and steady steam. Whisk in the milk. Bring to a boil, reduce to a bare simmer, then add the broccoli, carrot, hot pepper sauce, and black pepper. Simmer (look for occasional gentle bubbles) for 30 minutes.

Using a slotted spoon, transfer the broccoli from the soup to a blender or food processor. Add a generous splash of soup, about 1/2 cup. (Don't fill the blender more than halfway. Work in batches if you have a small blender.) Blend until the broccoli is almost smooth but not a paste.

Return the broccoli to the soup. Stir to combine. Add the Velveeta. Using a wooden spoon, stir until the Velveeta is melted (see note above). Season to taste with salt and simmer for 5 minutes. Serve. Store leftover soup in a covered container in the refrigerator for up to 3 days.